A natural generalisation of the model of random quadrangulations is that of Boltzmann random maps which allows the face degrees to be random, introduced in [1]}; it also naturally appears when considering random maps coupled with statistical physics models, see e.g. [2]}. Specifically, denote by \(\mathbf {M}\)  the set of all finite rooted bipartite planar maps, in which all faces have even degree. Let us fix a sequence of nonnegative real numbers \({q}= (q_i ; i \geqslant 1)\)  which, in order to avoid trivialities, satisfies \(q_i > 0\)  for at least one \(i \geqslant 2\) . We define a measure \(w^{{q}}\)  on \(\mathbf {M}\)  by setting
\(w^{{q}}(M) = \prod _{f \text{ face}} q_{\mathrm {deg}(f)/2},\qquad M \in \mathbf {M},\) 
